In high-speed dual-polarization MW-MC radio communication links adaptive XPIC algorithms are utilized to compensate cross-polarization interference between a vertical-polarized component and a horizontal-polarized component of a dual-polarized received radio signal.
In the general case of signal propagation in a dual-polarization MW-MC terrestrial radio communication link an elementary two-input-two-output (2×2) subband XPI canceller may be used. An n-th input sample of such canceller may be represented as a 2-vector containing both the vertical (V) and horizontal (H) polarization complex component:
                                          x            ⁡                          (              n              )                                =                                                    Hs                ⁡                                  (                  n                  )                                            +                              n                ⁡                                  (                  n                  )                                                      =                                          [                                                                                                                              x                          V                                                ⁡                                                  (                          n                          )                                                                                                                                                                                                  x                          H                                                ⁡                                                  (                          n                          )                                                                                                                    ]                            =                                                                    [                                                                                                                        h                            11                                                                                                                                h                            12                                                                                                                                                                            h                            21                                                                                                                                h                            22                                                                                                                ]                                    ⁡                                      [                                                                                                                                                      s                              V                                                        ⁡                                                          (                              n                              )                                                                                                                                                                                                                                      s                              H                                                        ⁡                                                          (                              n                              )                                                                                                                                            ]                                                  +                                  [                                                                                                                                          n                            V                                                    ⁡                                                      (                            n                            )                                                                                                                                                                                                                    n                            H                                                    ⁡                                                      (                            n                            )                                                                                                                                ]                                                                    ,                  n          =          1                ,        2        ,        …                            (        1        )            where sV(n) and sH(n) is the V- and H-polarization channel signal, correspondingly, nV(n) and nH(n) denote respectively the V- and H-polarization samples of stationary additive white Gaussian noise (AWGN), and H is the 2×2 channel matrix whose entries are unknown a priori. A channel model 100 according to equation (1) is depicted in FIG. 1.
The structure of the generalized 2×2 linear XPI canceller 200 is presented in FIG. 2 that depicts a nonsingular frequency-domain linear filter targeted to perform full separation of the V- and H-polarization channel signals over the corresponding frequency subband. A first subblock 201 processes the V-polarization channels and a second subblock 202 processes the H-polarization channels.
The canceller 200 is described by the following equation
                                          y            ⁡                          (              n              )                                =                                                    W                H                            ⁢                              x                ⁡                                  (                  n                  )                                                      =                                          [                                                                                                                              y                          V                                                ⁡                                                  (                          n                          )                                                                                                                                                                                                  y                          H                                                ⁡                                                  (                          n                          )                                                                                                                    ]                            =                                                [                                                                                                              w                          11                          *                                                                                                                      w                          21                          *                                                                                                                                                              w                          12                          *                                                                                                                      w                          22                          *                                                                                                      ]                                ⁡                                  [                                                                                                                                          x                            V                                                    ⁡                                                      (                            n                            )                                                                                                                                                                                                                    x                            H                                                    ⁡                                                      (                            n                            )                                                                                                                                ]                                                                    ,                                  ⁢                  n          =          1                ,        2        ,        …                            (        2        )            where x(n) and y(n) is the input and output signal vector sequence, correspondingly, W=[wV wH], and ( )H denotes Hermitian conjugation.
Under conditions of uncertainty about parameters of the two symbol streams, {sV(n)} and {sH(n)}, channel polarization matrix H entries, and variations of AWGN in equation (1), the problem of XPI cancellation can be solved by optimization of weight matrix W with respect to a suitable criterion. The methods developed basing on the adaptive approach applied to disclose the uncertainties can be roughly classified under the blind adaptive and adaptive trained categories. Because the blind adaptive algorithms require enormous volumes of input samples to converge they are considered to be not eligible for application in high-speed MW radio links.
For the adaptive trained XPIC problem, the weight matrix W can be found as a solution to the system of the minimum mean square error (MMSE) criteria:
                    {                                                                                                  min                                          w                      V                                                        ⁢                                      E                    ⁢                                          {                                                                                                                                                            ɛ                              V                                                        ⁡                                                          (                              n                              )                                                                                                                                2                                            }                                                                      =                                                      min                                          w                      V                                                        ⁢                                      E                    ⁢                                          {                                                                                                                                                                                            s                                V                                                            ⁡                                                              (                                n                                )                                                                                      -                                                                                          w                                V                                H                                                            ⁢                                                              x                                ⁡                                                                  (                                  n                                  )                                                                                                                                                                                              2                                            }                                                                                                                                                                                    min                                          w                      H                                                        ⁢                                      E                    ⁢                                          {                                                                                                                                                            ɛ                              H                                                        ⁡                                                          (                              n                              )                                                                                                                                2                                            }                                                                      =                                                      min                                          w                      H                                                        ⁢                                      E                    ⁢                                          {                                                                                                                                                                                            s                                H                                                            ⁡                                                              (                                n                                )                                                                                      -                                                                                          w                                H                                H                                                            ⁢                                                              x                                ⁡                                                                  (                                  n                                  )                                                                                                                                                                                              2                                            }                                                                                                                              (        3        )            
The weight vectors that minimize the corresponding mean square error (MSE), E{∥εV(n)∥2} and E{∥εH(n)∥2}, are
                    {                                                                                                  w                    Vopt                                    =                                                            c                      V                                        ⁢                                          R                      X                                              -                        1                                                              ⁢                                          v                      V                                                                                                                                                                w                    Hopt                                    =                                                            c                      H                                        ⁢                                          R                      X                                              -                        1                                                              ⁢                                          v                      H                                                                                                    ⁢                                          ⁢          where                                    (        4        )                                {                                                                              v                  V                                =                                  E                  ⁢                                      {                                                                  x                        ⁡                                                  (                          n                          )                                                                    ⁢                                                                        s                          V                          *                                                ⁡                                                  (                          n                          )                                                                                      }                                                                                                                                            v                  H                                =                                  E                  ⁢                                      {                                                                  x                        ⁡                                                  (                          n                          )                                                                    ⁢                                                                        s                          H                          *                                                ⁡                                                  (                          n                          )                                                                                      }                                                                                                          (        5        )            
sV(n), sH(n) are training signals, and cV, cH are scaling factors.
Since most high-speed dual-polarization MW-MC radio communication systems are designed to have very short training intervals, efficient algorithms of adaptive XPIC are required to have extremely fast convergence rate.
To be utilized in high-speed dual-polarization MW-MC radio communication links, an adaptive XPIC algorithm should have a maximum convergence rate that is a minimum number of training samples consumed to achieve the required gain of XPI suppression. The convergence rate of an eligible adaptive algorithm should be independent of signal propagation conditions and parameters of cross-polarization interference, particularly of the channel matrix structure. In order to have the shortest time of actual adaptation, an eligible XPIC algorithm should be capable of implementation on faster ASIC-based fixed-point processors with standard data/operation formats.
A wide variety of simple trained or supervised adaptive searching algorithms can be roughly classified under the gradient-based least-mean-squares (LMS) and boot-strapped sub-category. An adaptive trained searching LMS algorithm that iteratively approaches the minimum MSE weight coefficients in equation (4) is described as:
                    {                                                                                                  w                    V                                    ⁡                                      (                                          n                      +                      1                                        )                                                  =                                                                            w                      V                                        ⁡                                          (                      n                      )                                                        +                                                            μ                      V                                        ⁢                                                                  ɛ                        V                                            ⁡                                              (                        n                        )                                                              ⁢                                          x                      ⁡                                              (                        n                        )                                                                                                                                                                                                          w                    H                                    ⁡                                      (                                          n                      +                      1                                        )                                                  =                                                                            w                      H                                        ⁡                                          (                      n                      )                                                        +                                                            μ                      H                                        ⁢                                                                  ɛ                        H                                            ⁡                                              (                        n                        )                                                              ⁢                                          x                      ⁡                                              (                        n                        )                                                                                                                                                    (        6        )            where εV(n) and εH(n) are the errors as defined in equation (3) and μV>0 and μH>0 are the step size factors. A schematic of the adaptive trained LMS algorithm 300 according to equation (6) is depicted in FIG. 3 where abbreviation WP stands for a weight processor.
The error-feedback LMS algorithms are known to be simple, tolerant to numerical errors and noise, and well-provided for implementation on real-time ASIC-based fixed-point processors. However, this gradient-based searching algorithm sub-category possesses a number of properties that throws serious obstacles in the way of its efficient application in adaptive trained XPI cancellers. Such disadvantages include a strong dependence of the algorithm's convergence rate upon parameters of cross-polarization interference. In practical cases, the number of training sample pairs required to obtain an acceptable gain of XPI suppression may vary from tens to hundreds. Disadvantages further include significant influence of the LMS algorithms on the main V- and H-channel subband responses caused by the adaptively controlled complex weights, w11 and w22; and sensitivity of both the algorithm's convergence performance and stability to spurious amplitude modulation of input signals.
The underlying idea of the adaptive trained bootstrap XPIC algorithms sub-category is to decorrelate output signals of a nonsingular 2×2 linear filter to achieve full separation of two different cross-interfering symbol streams by deep suppression of cross-polarized signal components. A block-diagram of a fast-convergent adaptive bootstrap (BS) algorithm 400 is presented in FIG. 4.
This algorithm adaptively updates the V- and H-polarization weights, w21 and w21 to null the output covariance E{yV(n)yH*(n)} according to the following iterative equations:
                    {                                                                                                  w                    21                                    ⁡                                      (                                          n                      +                      1                                        )                                                  =                                                                            w                      21                                        ⁡                                          (                      n                      )                                                        -                                                                                                              η                          V                                                ⁡                                                  [                                                                                                                    y                                V                                                            ⁡                                                              (                                n                                )                                                                                      -                                                                                          s                                V                                                            ⁡                                                              (                                n                                )                                                                                                              ]                                                                    *                                        ⁢                                                                  y                        H                                            ⁡                                              (                        n                        )                                                                                                                                                                                                          w                    12                                    ⁡                                      (                                          n                      +                      1                                        )                                                  =                                                                            w                      12                                        ⁡                                          (                      n                      )                                                        -                                                                                                              η                          H                                                ⁡                                                  [                                                                                                                    y                                H                                                            ⁡                                                              (                                n                                )                                                                                      -                                                                                          s                                H                                                            ⁡                                                              (                                n                                )                                                                                                              ]                                                                    *                                        ⁢                                                                  y                        V                                            ⁡                                              (                        n                        )                                                                                                                                                    (        7        )            where ηV>0 and ηH>0 are the step size factors. The supervised adaptive BS algorithm 400 is sometimes faster and simpler for implementation than the LMS algorithms. However, a detailed analysis of the adaptive bootstrapped algorithm performance and simulation results has shown that the convergence rate of the bootstrapped algorithms remains dependent upon the cross-polarization parameters. In practical cases, the number of training sample pairs required to obtain an acceptable gain of XPI suppression still counts tens to hundreds. Because of operation of raising to the third power, the adaptive bootstrapped algorithms span a significantly wider dynamic range of arithmetic operations that requires extension of arithmetic precision. The bootstrapped algorithm's convergence performance and stability remain sensitive to spurious amplitude modulation of input signals.